ERBIL, Iraq — Iraq opened the next chapter in its offensive to drive the Islamic State out of Mosul on Sunday, preparing an assault on the western half of the city. Overnight, planes carpeted the ground with leaflets, directly appealing to the group’s fighters to surrender.

“To those of you who were intrigued by the ISIS ideology,” one of the leaflets said, “this is your last opportunity to quit your work with ISIS and to leave those foreigners who are in your homeland. Stay at home, raising the white flags as the forces approach.”

On state-run television, Prime Minister Haider al-Abadi announced the beginning of the offensive, describing it as “a new dawn” and calling on his troops “to move bravely forward to liberate what is left of the city.”

The Islamic State has held Mosul for nearly three years, and it is the group’s most important stronghold in Iraq. Retaking the entire city, in what is expected to be a monthslong campaign, would be a huge victory for Iraqi troops.

The assault is taking place amid concerns about the condition of hundreds of thousands of civilians trapped in the western part of the city. Food, water and cooking fuel have all been reported to be in short supply, and residents have described increased harassment from Islamic State fighters preparing for the attack.

The overall push to free Mosul, once Iraq’s second-largest city, began in October, with local troops pushing from the east into the city’s geographically larger but more sparsely populated eastern half. In late January, they reached the banks of the Tigris River, which bisects Mosul, and declared the city’s eastern section liberated.

The operation took longer than expected and took a high toll on civilians and the Iraqi forces, but much of the city’s infrastructure was preserved, and a sense of daily life has returned. That is in contrast to the operations to take back other cities, including Ramadi and Sinjar, which were decimated by airstrikes. More than a year after Sinjar was freed, even its mayor has not been able to return.

The fight for Mosul’s western half could be even more protracted than the fight for its east. The west is home to neighborhoods of narrow streets, some so small that it will not be possible for Iraqi troops to enter in their fortified Humvees. That may make the Islamic State’s signature suicide bomb attacks even more effective.

Because all five of the bridges spanning the Tigris have been bombed, Iraqi troops will trace a circuitous path to western Mosul, initially approaching it from the south.

Officials said the first objective would be Mosul International Airport, just south of the city. By midday on Sunday, Iraqi forces had captured a string of nearby villages and advanced within six miles of the airfield, officers said.

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American forces are supporting the operation. “The U.S. forces continue in the same role as they did in east Mosul,” Defense Secretary Jim Mattis told reporters traveling with him on Sunday, adding that the rules of engagement for American troops in Iraq had not changed. “We are very close to, if not already engaged in, that fight,” he said.

Mr. Mattis added that the American-backed coalition fighting the Islamic State would “continue with the accelerated effort to destroy” the group.

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While the airport may be unusable, taking it would still be a milestone for the offensive, as would taking the adjacent hilltop village of Abu Saif, which sits at a higher elevation than Mosul. Because of the Islamic State’s heavy use of snipers, securing high ground is crucial, and Iraqi forces were nearing the base of the hill by Sunday afternoon.

The troops’ push into western Mosul will be further complicated by the Islamic State’s vast network of tunnels throughout the city, which allow fighters to hide from overhead surveillance. And the group is increasingly using armed drones to spot and remotely bomb advancing Iraqi troops.

Yahya Salah, whose neighborhood in eastern Mosul was liberated in November, described how Iraqi troops were just streets away when Islamic State fighters forced their way into his home, armed with a jackhammer. They herded Mr. Salah’s family into one of the bedrooms. From behind the closed door, Mr. Salah said, he heard a deafening sound and realized the fighters were drilling a hole.

“They worked without stopping — when one got tired, another took over, and they dug a hole that was 1.5 meters wide,” said Mr. Salah, who said his family had been locked in the bedroom for three days. “When we said we were thirsty, they threw water bottles at us.”

He said the fighters had left at noon on the final day. The Iraqi Army arrived at sunset and unlocked the door. When the family stepped into the rest of their house, they found ceiling-high piles of dirt in three of their four bedrooms and a hole in the living room floor. The tunnel the fighters had dug stretched for dozens of yards, allowing the terrorist group’s foot soldiers to slip away.

Residents have shown reporters similar tunnels throughout the eastern part of the city, and officials expect the same in western Mosul. A photo essay published this weekend by the Islamic State, titled “Life of Fighters South of Mosul,” shows militants cooking a meal on a kerosene stove, reading the Quran and praying inside a tunnel wide enough for five men to stand side by side.

At the same time, the Islamic State has become better at the use of small drones, which are available off the shelf in malls across the region, including in Erbil, the nearest major city to Mosul. They use the drones to pinpoint army positions and to target them, and recently recovered Islamic State documents show how the group has cobbled together its own drone program. Iraqi forces say they frequently see the aircraft, two to four feet long, overhead, whining like lawn mowers. Then, 30 minutes later, they take incoming fire at that location.

“Mosul would be a tough fight for any army in the world, and the Iraqi forces have risen to the challenge,” Lt. Gen. Stephen Townsend, the commander of the American-led effort against the Islamic State, said in a news release from United States Central Command announcing the beginning of the operation. Some of the 450 American advisers on the ground in Iraq are helping Iraqi officers plan and carry out the offensive.

Reached by telephone, residents in western Mosul described the elation they felt at the approach of government troops. “All we have left to eat is tomato paste. We are eating it with salt,” said Umm Anwar, 41, who asked to be identified only by her nickname. “We are ready to kill ISIS ourselves with knives, or by biting them, because we are in so much pain.”

Rukmini Callimachi reported from Erbil, and Falih Hassan from Baghdad. Reporting was contributed by Omar Al-Jawoshy from Erbil; an employee of The New York Times from Mosul, Iraq; Helene Cooper from Abu Dhabi, United Arab Emirates; and Eric Schmitt from Washington.

A version of this article appears in print on February 20, 2017, on Page A1 of the New York edition with the headline: Iraq Opens Assault Against ISIS in West Mosul, With U.S. Help. Order Reprints|Today's Paper|Subscribe